CD4_CD8_Compare

library(Seurat)

## Attaching SeuratObject

library(SeuratData)

## ── Installed datasets ───────────────────────────────────── SeuratData v0.2.1 ──

## ✔ bmcite       0.3.0                    ✔ pbmcMultiome 0.1.0

## ────────────────────────────────────── Key ─────────────────────────────────────

## ✔ Dataset loaded successfully
## ❯ Dataset built with a newer version of Seurat than installed
## ❓ Unknown version of Seurat installed

library(cowplot)
library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

library(SAVER)
library(hexbin)
library(RColorBrewer)

load("CD4_CD8_dat.rds")
load("CD4_compare_corr.rds")
load("CD8_compare_corr.rds")

gene_names <- rownames(CD4_compare[[1]])

hex_vis <- function(sub_dat1, sub_dat2, idx1, idx2, title){
  bin1 <- hexbin(sub_dat1[, idx1], sub_dat1[, idx2], xbins = 20)
  bin2 <- hexbin(sub_dat2[, idx1], sub_dat2[, idx2], xbins = 20)
  my_colors <- colorRampPalette(rev(brewer.pal(11,'Spectral')))
  
  plot(bin1, colramp = my_colors,
       xlab = paste0(gene_names[idx1],
                     ", (", idx1, ")"),
       ylab = paste0(gene_names[idx2],
                     ", (", idx2, ")"),
       main = title[1])
  plot(bin2, colramp = my_colors,
       xlab = paste0(gene_names[idx1],
                     ", (", idx1, ")"),
       ylab = paste0(gene_names[idx2],
                     ", (", idx2, ")"),
       main = title[2])
}

1. Pearson Correlation

• Pearson’s correlation is to measure linear dependency of data, X and Y
• \(-1 \leq \rho_{Pearson}(X, Y) \leq 1\)
• \(\rho_{Pearson}(X, Y) = \frac{\sum(x_i-\bar{x})(y_i -\bar{y})}{\sum(x_i-\bar{x})^2(y_i -\bar{y})^2}\)

CD4_pear <- CD4_compare[[1]]; CD8_pear <- CD8_compare[[1]]

CD4_pear_quan <- quantile(abs(CD4_pear),
                          probs = c(0.05, 0.95),
                          na.rm = TRUE)

CD8_pear_quan <- quantile(abs(CD8_pear),
                          probs = c(0.05, 0.95),
                          na.rm = TRUE)

Low CD4 & High CD8

cor_contrast1 <- (CD4_pear < CD4_pear_quan[[1]]) & (CD8_pear > CD8_pear_quan[[2]])

cor_contrast_ind1 <- which(cor_contrast1, arr.ind = T)

nrow(cor_contrast_ind1)

## [1] 373

random_ind1 <- sample(nrow(cor_contrast_ind1), 5)
for (i in random_ind1){
  idx1 <- cor_contrast_ind1[i, 1]; idx2 <- cor_contrast_ind1[i, 2]
  title <- c(paste0("Pearson (CD4) of ", round(CD4_pear[idx1, idx2], 3)),
             paste0("Pearson (CD8) of ", round(CD8_pear[idx1, idx2], 3)))
  hex_vis(CD4_subdat, CD8_subdat, idx1, idx2, title)
}

High CD4 & Low CD8

cor_contrast2 <- (CD4_pear > CD4_pear_quan[[2]]) & (CD8_pear < CD8_pear_quan[[1]])

cor_contrast_ind2 <- which(cor_contrast2, arr.ind = T)

nrow(cor_contrast_ind2)

## [1] 414

random_ind2 <- sample(nrow(cor_contrast_ind2), 5)

for (i in random_ind2){
  idx1 <- cor_contrast_ind2[i, 1]; idx2 <- cor_contrast_ind2[i, 2]
  title <- c(paste0("Pearson (CD4) of ", round(CD4_pear[idx1, idx2], 3)),
             paste0("Pearson (CD8) of ", round(CD8_pear[idx1, idx2], 3)))
  hex_vis(CD4_subdat, CD8_subdat, idx1, idx2, title)
}

2. Spearman Correlation

• It captures the monotonic relationship between data, X and Y
• \(-1 \leq \rho_{Spearman}(X,Y) \leq 1\)
• \(\rho_{Spearman} = 1 - \frac{6\sum{d_i^2}}{n(n^2-1)}\) where \(d_i\) is the difference between the ranks of \(x_i\) and \(y_i\)

CD4_spear <- CD4_compare[[2]]; CD8_spear <- CD8_compare[[2]]

CD4_spear_quan <- quantile(abs(CD4_spear),
                           probs = c(0.10, 0.90),
                           na.rm = TRUE)

CD8_spear_quan <- quantile(abs(CD8_spear),
                           probs = c(0.10, 0.90),
                           na.rm = TRUE)

Low CD4 & High CD8

cor_contrast3 <- (CD4_spear < CD4_spear_quan[[1]]) & (CD8_spear > CD8_spear_quan[[2]])

cor_contrast_ind3 <- which(cor_contrast3, arr.ind = T)

nrow(cor_contrast_ind3)

## [1] 6

random_ind3 <- sample(nrow(cor_contrast_ind3), 5)
for (i in random_ind3){
  idx1 <- cor_contrast_ind3[i, 1]; idx2 <- cor_contrast_ind3[i, 2]
  title <- c(paste0("Spearman (CD4) of ", round(CD4_spear[idx1, idx2], 3)),
             paste0("Spearman (CD8) of ", round(CD8_spear[idx1, idx2], 3)))
  hex_vis(CD4_subdat, CD8_subdat, idx1, idx2, title)
}

High CD4 & Low CD8

cor_contrast4 <- (CD4_spear > CD4_spear_quan[[2]]) & (CD8_spear < CD8_spear_quan[[1]])

cor_contrast_ind4 <- which(cor_contrast4, arr.ind = T)

nrow(cor_contrast_ind4)

## [1] 303

random_ind4 <- sample(nrow(cor_contrast_ind4), 5)

for (i in random_ind4){
  idx1 <- cor_contrast_ind4[i, 1]; idx2 <- cor_contrast_ind4[i, 2]
  title <- c(paste0("Spearman (CD4) of ", round(CD4_spear[idx1, idx2], 3)),
             paste0("Spearman (CD8) of ", round(CD8_spear[idx1, idx2], 3)))
  hex_vis(CD4_subdat, CD8_subdat, idx1, idx2, title)
}

3. Kendall’s Tau

• It is an alternative method to Spearman’s correlations, identifying monotonic relationships.
• \(-1 \leq \rho_{Kendall}(X,Y) \leq 1\)
• \(\rho_{Kendall}(X,Y) = \frac{\#\;concordant\;pairs - \#\;discordant \;pairs}{0.5n(n-1)}\)

CD4_kendall <- CD4_compare[[3]]; CD8_kendall <- CD8_compare[[3]]

CD4_kendall_quan <- quantile(abs(CD4_kendall),
                             probs = c(0.10, 0.90),
                             na.rm = TRUE)

CD8_kendall_quan <- quantile(abs(CD8_kendall),
                             probs = c(0.10, 0.90),
                             na.rm = TRUE)

Low CD4 & High CD8

cor_contrast5 <- (CD4_kendall < CD4_kendall_quan[[1]]) & (CD8_kendall > CD8_kendall_quan[[2]])

cor_contrast_ind5 <- which(cor_contrast5, arr.ind = T)

nrow(cor_contrast_ind5)

## [1] 6

random_ind5 <- sample(nrow(cor_contrast_ind5), 5)
for (i in random_ind5){
  idx1 <- cor_contrast_ind5[i, 1]; idx2 <- cor_contrast_ind5[i, 2]
  title <- c(paste0("Kendall (CD4) of ", round(CD4_kendall[idx1, idx2], 3)),
             paste0("Kendall (CD8) of ", round(CD8_kendall[idx1, idx2], 3)))
  hex_vis(CD4_subdat, CD8_subdat, idx1, idx2, title)
}

High CD4 & Low CD8

cor_contrast6 <- (CD4_kendall > CD4_kendall_quan[[2]]) & (CD8_kendall < CD8_kendall_quan[[1]])

cor_contrast_ind6 <- which(cor_contrast6, arr.ind = T)

nrow(cor_contrast_ind6)

## [1] 321

random_ind6 <- sample(nrow(cor_contrast_ind6), 5)

for (i in random_ind6){
  idx1 <- cor_contrast_ind6[i, 1]; idx2 <- cor_contrast_ind6[i, 2]
  title <- c(paste0("Kendall (CD4) of ", round(CD4_kendall[idx1, idx2], 3)),
             paste0("Kendall (CD8) of ", round(CD8_kendall[idx1, idx2], 3)))
  hex_vis(CD4_subdat, CD8_subdat, idx1, idx2, title)
}

6. Normalized Mutual Information

• It measures how much one random variable gives information about the other. For example, High mutual information indicates a large reduction in uncertainty.
• \(0 \leq MI(X,Y) \leq 1\), as it is normalized.
• $MI(X,Y) = f_{X,Y} (x,y) log_2 ; dxdy $
• \(MI(X,Y) = \sum \sum p_{X,Y} (x,y) log \frac{p_{X,Y} (x,y)}{P_X(x)P_Y(y)}\)

CD4_NMI <- CD4_compare[[6]]; CD8_NMI <- CD8_compare[[6]]

CD4_NMI_quan <- quantile(abs(CD4_NMI),
                         probs = c(0.025, 0.975),
                         na.rm = TRUE)

CD8_NMI_quan <- quantile(abs(CD8_NMI),
                         probs = c(0.025, 0.975),
                         na.rm = TRUE)

Low CD4 & High CD8

cor_contrast9 <- (CD4_NMI < CD4_NMI_quan[[1]]) & (CD8_NMI > CD8_NMI_quan[[2]])

cor_contrast_ind9 <- which(cor_contrast9, arr.ind = T)

nrow(cor_contrast_ind9)

## [1] 1495

random_ind9 <- sample(nrow(cor_contrast_ind9), 5)
for (i in random_ind9){
  idx1 <- cor_contrast_ind9[i, 1]; idx2 <- cor_contrast_ind9[i, 2]
  title <- c(paste0("NMI (CD4) of ", round(CD4_NMI[idx1, idx2], 3)),
             paste0("NMI (CD8) of ", round(CD8_NMI[idx1, idx2], 3)))
  hex_vis(CD4_subdat, CD8_subdat, idx1, idx2, title)
}

High CD4 & Low CD8

cor_contrast10 <- (CD4_NMI > CD4_NMI_quan[[2]]) & (CD8_NMI < CD8_NMI_quan[[1]])

cor_contrast_ind10 <- which(cor_contrast10, arr.ind = T)

nrow(cor_contrast_ind10)

## [1] 186

random_ind10 <- sample(nrow(cor_contrast_ind10), 5)

for (i in random_ind10){
  idx1 <- cor_contrast_ind10[i, 1]; idx2 <- cor_contrast_ind10[i, 2]
  title <- c(paste0("NMI (CD4) of ", round(CD4_NMI[idx1, idx2], 3)),
             paste0("NMI (CD8) of ", round(CD8_NMI[idx1, idx2], 3)))
  hex_vis(CD4_subdat, CD8_subdat, idx1, idx2, title)
}

7. Distance Correlation

• it is measure to identify non-linear relationships between two random variables with energy distances.
• distance correlation is calculated by dividing the distance covariance between X and Y by the product of their distance standard deviations.
• \(0 \leq dCor \leq 1\)
• \(dCor(X,Y) = \frac{dCov(Y,Y)}{\sqrt{dVar(X)dVar(Y)}}\)
  • \(dCov(X, Y) = \sqrt{\frac{1}{n^2} \sum_{k=1, l=1}^{n} A_{k,l}B_{k,l}}\)
  • \(dVar(X) = dCov(X,X) and dVar(Y) = dCov(Y, Y)\)

CD4_dist <- CD4_compare[[7]]; CD8_dist <- CD8_compare[[7]]

CD4_dist_quan <- quantile(abs(CD4_dist),
                          probs = c(0.05, 0.95),
                          na.rm = TRUE)

CD8_dist_quan <- quantile(abs(CD8_dist),
                          probs = c(0.05, 0.95),
                          na.rm = TRUE)

Low CD4 & High CD8

cor_contrast11 <- (CD4_dist < CD4_dist_quan[[1]]) & (CD8_dist > CD8_dist_quan[[2]])

cor_contrast_ind11 <- which(cor_contrast11, arr.ind = T)

nrow(cor_contrast_ind11)

## [1] 5

random_ind11 <- sample(nrow(cor_contrast_ind11), 5)

for (i in random_ind11){
  idx1 <- cor_contrast_ind11[i, 1]; idx2 <- cor_contrast_ind11[i, 2]
  title <- c(paste0("Dist (CD4) of ", round(CD4_dist[idx1, idx2], 3)),
             paste0("Dist (CD8) of ", round(CD8_dist[idx1, idx2], 3)))
  hex_vis(CD4_subdat, CD8_subdat, idx1, idx2, title)
}

High CD4 & Low CD8

cor_contrast12 <- (CD4_dist > CD4_dist_quan[[2]]) & (CD8_dist < CD8_dist_quan[[1]])

cor_contrast_ind12 <- which(cor_contrast12, arr.ind = T)

nrow(cor_contrast_ind12)

## [1] 35

random_ind12 <- sample(nrow(cor_contrast_ind12), 5)

for (i in random_ind12){
  idx1 <- cor_contrast_ind12[i, 1]; idx2 <- cor_contrast_ind12[i, 2]
  title <- c(paste0("Dist (CD4) of ", round(CD4_dist[idx1, idx2], 3)),
             paste0("Dist (CD8) of ", round(CD8_dist[idx1, idx2], 3)))
  hex_vis(CD4_subdat, CD8_subdat, idx1, idx2, title)
}

8. Chatterjee’s XI Correlation

• It measures the degree of dependence between the variables with concept of rank.
• \(0 \leq \xi_n \leq 1\)
• \(\xi_n(X,Y) = 1 - \frac{3\sum_{i=1}^{n-1} |r_{i+1} -r_i|}{n^2-1}\)
• \(\xi_n(X,Y) = 1 - \frac{n\sum_{i=1}^{n-1}|r_{i+1}-r_i}{2\sum_{i=1}^{n}l_i(n-l_i)}\)

CD4_XI <- CD4_compare[[8]]; CD8_XI <- CD8_compare[[8]]

CD4_XI_quan <- quantile(abs(CD4_XI),
                        probs = c(0.025, 0.975),
                        na.rm = TRUE)

CD8_XI_quan <- quantile(abs(CD8_XI),
                        probs = c(0.025, 0.975),
                        na.rm = TRUE)

Low CD4 & High CD8

cor_contrast13 <- (CD4_XI < CD4_pear_quan[[1]]) & (CD8_pear > CD8_pear_quan[[2]])

cor_contrast_ind13 <- which(cor_contrast13, arr.ind = T)

nrow(cor_contrast_ind13)

## [1] 28

random_ind13 <- sample(nrow(cor_contrast_ind13), 5)

for (i in random_ind13){
  idx1 <- cor_contrast_ind13[i, 1]; idx2 <- cor_contrast_ind13[i, 2]
  title <- c(paste0("XI (CD4) of ", round(CD4_XI[idx1, idx2], 3)),
             paste0("XI (CD8) of ", round(CD8_XI[idx1, idx2], 3)))
  hex_vis(CD4_subdat, CD8_subdat, idx1, idx2, title)
}

High CD4 & Low CD8

cor_contrast14 <- (CD4_XI > CD4_pear_quan[[2]]) & (CD8_XI < CD8_pear_quan[[1]])

cor_contrast_ind14 <- which(cor_contrast14, arr.ind = T)

nrow(cor_contrast_ind14)

## [1] 3254

random_ind14 <- sample(nrow(cor_contrast_ind14), 5)

for (i in random_ind14){
  idx1 <- cor_contrast_ind14[i, 1]; idx2 <- cor_contrast_ind14[i, 2]
  title <- c(paste0("XI (CD4) of ", round(CD4_XI[idx1, idx2], 3)),
             paste0("XI (CD8) of ", round(CD8_XI[idx1, idx2], 3)))
  hex_vis(CD4_subdat, CD8_subdat, idx1, idx2, title)
}